I came across this question which asks me to draw the graph by hand(just with hand and head) for the following curve:-
$$x^5+y^5=5a^2xy^2$$
Here, I don't think swapping is going to work out. So here is what I usually do while drawing curves.
#Points of Intersection with the Cordinate Axes
Here there is only one point that of origin, thus the curves passes through origin.
#Finding any asymptotes
Here $x+y=0$ is the only asymptote for the curve
#Odd/even/Neither Odd nor even function
Here $x \rightarrow -x, y \rightarrow -y$ Gives us back the original curve thus the curves is symmetrical in opposite quadrants.
#Monotonicity of intervals
But here is where my problem arises, I am not able to evaluate the curve’s derivative.
#Evaluating concavity/convexity
If I am not able to evaluate the first derivative then I am not going to be able to get till here. Moreover the last two parts are the most crucial and without being able to evaluate them, I will not be able to draw the curve.
So my question is can we find the derivative just in terms of $x$ somehow? If not, then is there some other method to this?
TL;DR Is there any way of drawing the curve $x^5+y^5=5a^2xy^2$ by hand?